Lesson 3.14

Holes

A hole (removable discontinuity) occurs when a factor cancels from both the numerator and denominator. The graph looks continuous — except for a single missing point.

Introduction

In Lesson 3.13, we saw that a zero in the denominator creates a vertical asymptote — unless the same factor also appears in the numerator and cancels. When it cancels, the discontinuity becomes a hole: a single missing point.

Past Knowledge

Vertical asymptotes (3.13) and simplifying rational expressions (3.2).

Today's Goal

Find holes by identifying cancelled factors, then compute the y-coordinate of the hole.

Future Success

Distinguishing holes from VAs is crucial for accurate graph sketching in Lesson 3.18.

Key Concepts

Finding a Hole

Factor numerator and denominator

Identify common factors that cancel

The cancelled factor gives the x-coordinate: set it = 0

Plug that x into the SIMPLIFIED function for the y-coordinate

The hole is the point (x, y)

VA vs. Hole Summary

Feature	Vertical Asymptote	Hole
Factor cancels?	No	Yes
Graph behavior	Goes to ±∞	Missing point
Notation	Dashed vertical line	Open circle ○

Worked Examples

Example 1: Simple Hole

Basic

Find any holes in .

cancels → hole at

Simplified:

y-coordinate:

Hole at , VA at

Example 2: Factor First

Intermediate

Find any holes in .

Factor numerator

cancels → hole at

Simplified:

Hole at . Graph looks like the line with one point missing.

Example 3: Hole + VA Together

Advanced

Classify the discontinuities of .

Factor both

cancels → hole at

Simplified: . y-val:

does NOT cancel → VA at

Hole: | VA:

Common Pitfalls

Using the Original Function for y

You can't plug the hole's x-value into the original — it gives . Use the simplified function.

Thinking Holes Are Visible

Holes are invisible on a graphing calculator. You must find them algebraically.

Real-Life Applications

In signal processing, removable discontinuities represent "glitches" that can be patched. Understanding holes helps engineers identify and correct these artifacts in digital data.